Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{9x^3 + 18x^2 - 720x}{9x^2 - 900}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {9x(x^2 + 2x - 80)} {9(x^2 - 100)} $ $ p = \dfrac{9x}{9} \cdot \dfrac{x^2 + 2x - 80}{x^2 - 100} $ Simplify: $ p = x \cdot \dfrac{x^2 + 2x - 80}{x^2 - 100}$ Next factor the numerator and denominator. $ p = x \cdot \dfrac{(x + 10)(x - 8)}{(x + 10)(x - 10)}$ Assuming $x \neq -10$ , we can cancel the $x + 10$ $ p = x \cdot \dfrac{x - 8}{x - 10}$ Therefore: $ p = \dfrac{ x(x - 8)}{ x - 10 }$, $x \neq -10$